Finding optimal k0 values

From Polymath1Wiki

(Difference between revisions)
Jump to: navigation, search
(Benchmarks)
(Code and data)
Line 509: Line 509:
== Code and data ==
== Code and data ==
-
* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding an optimal <math> k_0</math>]: Java, bash, and maple are needed.
+
* [http://www.cs.cmu.edu/~xfxie/software/K0Finder.zip Code for finding optimal <math> k_0</math> values] using a simple optimizer [http://www.cs.cmu.edu/~xfxie/software/depso.zip DEPSO]: Java, bash, and maple are needed.

Revision as of 18:36, 20 October 2013

This is a sub-page for the Polymath8 project "bounded gaps between primes".

  • ~k_0~ (~k_0 \ge 2, k_0 \in \Z~) is a quantity such that every admissible ~k_0-tuple has infinitely many translates which each contain at least two primes. It is then used for finding narrow admissible tuples.
  • \text{MPZ}^{(i)}[\varpi,\delta] holds for some combinations of c_\varpi, c_\delta, and ~i~ values, where i \ge 1 means ~i-tuply densely divisible, c_\varpi > 0 and ~c_\delta > 0~ are constants in the constraint on \varpi and ~\delta~, such that c_{\varpi}\varpi+c_{\delta}\delta<1.

Contents

Optimization Model

For a given set of c_\varpi, c_\delta, i, the basic optimization model is defined as:


\text{minimize}~~k_0~


\text{subject to:}~

2(\kappa_1+\kappa_2+\kappa_3) < \left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right)
c_\varpi \varpi + c_\delta \delta < 1,
0 < \varpi <1/4,
0 < \delta \leq \delta' < \frac{1}{4} + \varpi,
A \ge 0,


\text{where}~

 \kappa_1 := \int_{\theta}^1 (1-t)^{(k_0-1)/2} \frac{dt}{t}
 \kappa_2 := (k_0-1) \int_{\theta}^1 (1-t)^{k_0-1} \frac{dt}{t}
 \kappa_3 := \tilde \theta \frac{J_{k_0-2}(\sqrt{\tilde \theta} j_{k_0-2})^2 - J_{k_0-3}(\sqrt{\tilde \theta} j_{k_0-2}) J_{k_0-1}(\sqrt{\tilde \theta} j_{k_0-2})}{ J_{k_0-3}(j_{k_0-2})^2 } 
\exp( A + (k_0-1) \int_{\tilde \delta}^\theta e^{-(A+2\alpha)t} \frac{dt}{t} )
 \alpha := \frac{j_{k_0-2}^2}{4(k_0-1)}
 \theta := \frac{\delta'}{1/4 + \varpi}
 \tilde \theta := \frac{(i\delta' - \delta)/2 + \varpi}{1/4 + \varpi}
 \tilde \delta := \frac{\delta}{1/4 + \varpi}

and \varpi, \delta, \delta', A are parameters to be optimized.

More details are described in the page on Dickson-Hardy-Littlewood theorems.

Benchmarks

The following table provides the results with clean parameter values for the best currently known instances of c_\varpi, c_\delta, i values, respectively without and with using Deligne's theorem.

Clean results at k_0 = k_0^{opt} for the best currently known instances of c_\varpi, c_\delta, i values.
Instance k_0^{*} ~k_0~ Parameters Error Terms Comment
c_{\varpi} ~c_{\delta}~ ~i~ \varpi ~\delta~ ~\delta'~ ~A~ ~\kappa_1~ ~\kappa_2~ ~\kappa_3~
168 48 2 1781 1783 5.950000E-03 1E-05 1/300 800 6.662E-07 5.209E-09 8.340E-47 Without Deligne's theorem
600/7 180/7 4 630 632 1.163666E-02 1E-04 1/105 200 6.445E-06 1.752E-08 7.018E-08 With Deligne's theorem

The following table provides the results with minimized parameter values at k_0 = k_0^{opt} for some instances of c_\varpi, c_\delta, i values.

Optimal results at k_0 = k_0^{opt} for some instances of c_\varpi, c_\delta, i values.
Instance k_0^{*} ~k_0~ Parameters Error Terms Objective
c_{\varpi} ~c_{\delta}~ ~i~ \varpi ~\delta~ ~\delta'~ ~A~ ~\kappa_1~ ~\kappa_2~ ~\kappa_3~
348 68 1 5446 5447 2.8733351E-03 1.1672627E-06 1.4961657E-03 2559.258877 5.59E-09 1.50E-12 6.02E-11 -1.1882E-06
168 48 2 1781 1783 5.9495534E-03 9.8965035E-06 3.7117059E-03 757.8242621 1.58E-07 3.24E-10 3.65E-09 -5.9684E-06
148 33 1 1465 1466 6.7542244E-03 1.1357314E-05 4.7101572E-03 626.6135921 8.79E-08 8.57E-11 3.63E-09 -2.2867E-06
140 32 1 1345 1346 7.1398444E-03 1.3180858E-05 5.0540952E-03 577.7849932 1.10E-07 1.22E-10 4.75E-09 -6.7812E-06
116 30 1 1006 1007 8.6150244E-03 2.1905745E-05 6.4310210E-03 408.9674914 2.29E-07 3.76E-10 1.20E-08 -6.2561E-06
108 30 1 901 902 9.2518776E-03 2.6573843E-05 7.0318847E-03 359.6376563 3.08E-07 6.00E-10 1.76E-08 -1.0924E-05
280/3 80/3 2 719 720 1.0699851E-02 5.0521044E-05 8.0398983E-03 260.2624368 1.04E-06 4.98E-09 4.33E-08 -5.5687E-06
600/7 180/7 4 630 632 1.1639206E-02 9.1536798E-05 8.3866560E-03 194.5246551 3.01E-06 3.40E-08 9.89E-08 -5.0940E-06

The following table provides the "unsatisfied" results with minimized parameter values at k_0=(k_0^{opt}-1) for some instances of c_\varpi, c_\delta, i values. Note that the results of these k0 values do not satisfy all constraints, as indicated from the positive objective values.

"Unsatisfied" optimal results at k_0=(k_0^{opt}-1) for some instances of c_\varpi, c_\delta, i values.
Instance k_0^{*} ~k_0~ Parameters Error Terms Objective
c_{\varpi} ~c_{\delta}~ ~i~ \varpi ~\delta~ ~\delta'~ ~A~ ~\kappa_1~ ~\kappa_2~ ~\kappa_3~
348 68 1 5446 5446 2.8733354E-03 1.1660200E-06 1.4880084E-03 2552.313151 6.16E-09 1.81E-12 1.00E-10 1.7550E-07
168 48 2 1781 1782 5.9495511E-03 9.9043741E-06 3.7130742E-03 757.3673135 1.59E-07 3.26E-10 3.21E-09 2.5064E-06
148 33 1 1465 1465 6.7542239E-03 1.1359571E-05 4.7002144E-03 625.1479808 9.16E-08 9.27E-11 3.28E-09 9.3639E-06
140 32 1 1345 1345 7.1398419E-03 1.3191801E-05 5.0550954E-03 567.8210511 1.11E-07 1.24E-10 4.65E-10 6.6029E-06
116 30 1 1006 1006 8.6150194E-03 2.1925014E-05 6.4287825E-03 408.5511082 2.33E-07 3.89E-10 1.24E-08 1.5183E-05
108 30 1 901 901 9.2518661E-03 2.6615167E-05 7.0404135E-03 359.5845846 3.08E-07 5.97E-10 1.68E-08 1.4703E-05
280/3 80/3 2 719 719 1.0699822E-02 5.0626919E-05 8.0520479E-03 259.8370595 1.04E-06 4.96E-09 4.46E-08 3.1365E-05
600/7 180/7 4 630 631 1.1639134E-02 9.1775130E-05 8.3989836E-03 193.9881059 3.02E-06 3.40E-08 1.00E-07 4.0614E-05

The following table provides the results with minimized parameter values at k_0=(k_0^{opt}-1) for some instances of c_\varpi, c_\delta, i values, with an additional condition that \left(1- \frac{j_{k_0-2}^2}{k_0(k_0-1)(1 + 4 \varpi)}\right) \ge 0. Note that these k0 values are more unsatisfied, as indicated from the larger positive objective values that that in the previous table.


More conservative "unsatisfied" results at k_0=(k_0^{opt}-1) for some instances of c_\varpi, c_\delta, i values.
Instance k_0^{*} ~k_0~ Parameters Error Terms Objective
c_{\varpi} ~c_{\delta}~ ~i~ \varpi ~\delta~ ~\delta'~ ~A~ ~\kappa_1~ ~\kappa_2~ ~\kappa_3~
168 48 2 1781 1782 5.9501100E-03 7.9483333E-06 1.9082658E-03 777.7015422 1.68E-04 2.02E-04 9.81E-06 7.6073E-04
600/7 180/7 4 630 631 1.1648112E-02 6.1848056E-05 4.2588144E-03 222.5549310 9.33E-04 1.79E-03 1.02E-04 5.6566E-03

Lower Bounds

For each ~c_\varpi, a theoretical lower bound of ~k_0, called k_0^*, can be obtained by assuming that all error terms ~\kappa_1, ~\kappa_2, and ~\kappa_3 could be completely ignored. This table gives the computational results of k_0^* for c_\varpi < 87 .


~c_\varpi~ 0 1 2 3 4 5 6 7 8 9
80 566 577 588 599 611 622 633 - - -
70 460 470 481 491 502 512 523 533 544 555
60 362 372 381 391 400 410 420 430 440 450
50 273 281 290 299 307 316 325 334 343 353
40 193 200 208 216 223 231 239 248 256 264
30 123 129 136 143 149 156 163 171 178 185
20 65 70 76 81 87 92 98 104 110 117
10 22 26 30 33 38 42 46 51 55 60
00 - - - - 6 8 10 13 16 19

Code and data

Personal tools